Question

answer part 2. * 4 (10 points)
complete the proof.
given: ( overline{bd} ) bisects ( overline{ac} ) at point ( e ; angle b a e cong angle d c e ).
prove: ( \triangle a b e cong \triangle c d e )
we are given that ( overline{bd} ) bisects ( overline{ac} ) at point ( e ) and ( angle b a e cong angle d c e ).
because ( overline{bd} ) bisects ( overline{ac} ) at point ( e, overline{ae} cong overline{ce} ) by the
a. definition of a segment bisector b. segment addition postulate.
by the vertical angles theorem, ( angle a e b cong angle c e d ). therefore, ( \triangle a b e cong \triangle c d e )
bv the a. aas theorem b. asa postulate c. sas postulate
sas postulate
aas theorem
asa postulate

Explanation:

Step1: Recall triangle congruence criteria

We have two angles and a non - included side (AAS) or two angles and the included side (ASA) or two sides and the included angle (SAS). Here, we know that \(AE = CE\) (given as \(BD\) bisects \(AC\)), \(\angle BAE=\angle DCE\) (given), and \(\angle AEB=\angle CED\) (vertical angles are congruent).

Step2: Apply the AAS Theorem

The AAS (Angle - Angle - Side) Theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. In \(\triangle ABE\) and \(\triangle CDE\), we have \(\angle BAE=\angle DCE\), \(\angle AEB=\angle CED\), and \(AE = CE\).

Answer:

AAS Theorem